Jeg trykker på knappen, og døren åbner.

They are both in the present tense.

• at trykke → trykker
• at åbne → åbner

In Danish, the present tense is usually made by adding -r to the infinitive.

So this sentence uses the same kind of tense as English I press ... the door opens.

No. Danish verbs do not change for person or number.

That means:

• jeg trykker = I press
• du trykker = you press
• han trykker = he presses
• de trykker = they press

The verb form stays the same. This is much simpler than English.

Because trykke på is the normal Danish expression for pressing a button.

So:

• trykke på knappen = press the button

The preposition på is part of the natural phrasing here. English speakers often want to translate word-for-word and leave it out, but in Danish på sounds right with buttons, switches, and similar things.

Because Danish usually adds the definite article to the end of the noun.

So:

• en knap = a button

• knappen = the button

• en dør = a door
• døren = the door

This -en ending is used with many common gender nouns (also called en-words).

It is built into the noun itself.

In this sentence:

• knappen already means the button
• døren already means the door

Danish often uses a suffix instead of a separate word like English the.

Yes. Danish can use åbne both:

• transitively: someone opens something
  
  • Jeg åbner døren = I open the door
• intransitively: something opens
  
  • Døren åbner = The door opens

So døren åbner is completely normal. It means the door opens by itself as the result of what happened before.

Åbnes is a passive form, meaning something like the door is opened.

• døren åbner = the door opens
• døren åbnes = the door is opened

In your sentence, døren åbner sounds more natural and direct. It focuses on what happens to the door, not on an unnamed agent.

Both parts are normal Danish main clauses, and Danish main clauses usually follow verb-second word order.

In practice, that often looks like:

• subject + verb + rest

So here you get:

• Jeg
  • trykker
    • på knappen
• døren
  • åbner

Because both clauses begin with the subject, the order looks very similar to English.

Also, after og in this sentence, the second clause keeps normal main-clause order:

• og døren åbner not
• og åbner døren

It separates the sentence into two clauses:

• Jeg trykker på knappen
• og døren åbner

So it helps show that one full action is followed by another.

In real Danish writing, punctuation around og can vary a bit depending on style, but the important grammar point for a learner is that these are two coordinated clauses joined by og.

It can do both, depending on context.

This sentence could mean:

• something happening right now: I press the button, and the door opens
• something that generally happens: When I press the button, the door opens

Danish present tense is often used in the same broad way as English present tense.

They are Danish vowels that do not have perfect English equivalents.

• ø in døren is pronounced with rounded lips. It is somewhat like the vowel in French deux or German schön.
• å in åbner is roughly like the vowel in English law or more, depending on accent.

And y in trykker is also special: it is a front vowel said with rounded lips, a bit like saying ee while rounding your lips.

These sounds usually take practice, because English does not use them in the same way.